Iman Setayesh's Generals, Spring 2007

Special topics: Algebraic Geometry, Algebraic Topology

Committee: Pandharipandhe(chair), Szabo, Stein

(2 hours)

Algebra

P:What can you say about matrices that satisfy a given polynomial (over an algebraically
closed field)? How many of them are there?(Jordan form)
P:What about non-alg. closed fields?(Rational form)
P:What are the prime ideals and maximal ideals of Z[x]?

Complex Analysis

St:What can you say about a holomorphic function with a given sequence of points as zeroes?
What can you say in special case of a_n=n?
St:can you write down the gamma funcion (integral definition)? How is it related to the
function of previous question?
St:Do you know any function with poles at a lattice?(I wrote down the Wirestrass's function)
Why did you subtract the 1/(w^2) terms? 
Sz:what is the maximum modulus theorem? How can you prove it?
Sz:State the open mapping theorem. State Rouche's theorem.

Real Analysis

St:What are the Hilbert spaces? Give us two hilbert spaces which aren't isomorphic in an
obvious way but there is an isomorphism between them!
(I cofused myself! so they started giving hints)Give an example of a Hilbert space.(L^2(R) )
what about S^1 and Z .(aha! Fourirer series) 
St:OK now that you mentioned it what do you know about fourier series? How is the convergent
for these series? What do you mean by a a basis for Hilbert space?
St:If we start with e^(2*pi*i*n) with n>=0 what would be the closur of the algebra generated
by them? State Stone-Wirestrass theorem( I stated it for real valued 
functions)what about complex valued functions?(conjugation condition)

Algebraic Geometry and Algebraic Toplogy

P:What can you say about a non_singular quadric in P^3?
-I can compute some of its cohomology groups!
P:OK go on. Try to copute the Hodge dimond of this surface.
(using 0->I->O(P^3)->O(X)->0, I computed all entries but the one in the middle) 
Sz:Can you compute it's euler characteristic?(exact sequence for normal bundle to get the top
chern class,...) 

Sz:suppose you have unknot and trifold knot compute the fundemental group of their complement
in R^3. State the Vankampen's theorem. Consider a neigbourhood 
of your knot,what is the image of fundemental group of this torus inside the fundamental group
of (R^3-knot)
Sz:What about homology groups of the complement? State Mayer_vitoris.

P:Given an integer g construct a curve of that genus. Can you always find a non_hyperellipti
 one?(g>2, dimension counting) For g=3 give an explicit example of a 
non-hyperelliptic curve of genus 3.(degree 4 curves in P^2)